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SageMath

sage: E = EllipticCurve("ca1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.ca

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|

405600.ca1 | 405600ca4 | \([0, -1, 0, -4399633, -3550512863]\) | \(30488290624/195\) | \(60238576320000000\) | \([2]\) | \(6193152\) | \(2.4044\) | |

405600.ca2 | 405600ca2 | \([0, -1, 0, -914008, 273851512]\) | \(2186875592/428415\) | \(16543019021880000000\) | \([2]\) | \(6193152\) | \(2.4044\) |
\(\Gamma_0(N)\)-optimal^{*} |

405600.ca3 | 405600ca1 | \([0, -1, 0, -280258, -53163488]\) | \(504358336/38025\) | \(183539412225000000\) | \([2, 2]\) | \(3096576\) | \(2.0578\) |
\(\Gamma_0(N)\)-optimal^{*} |

405600.ca4 | 405600ca3 | \([0, -1, 0, 268992, -236612988]\) | \(55742968/658125\) | \(-25413149385000000000\) | \([2]\) | \(6193152\) | \(2.4044\) |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405600.ca1.

## Rank

sage: E.rank()

The elliptic curves in class 405600.ca have rank \(0\).

## Complex multiplication

The elliptic curves in class 405600.ca do not have complex multiplication.## Modular form 405600.2.a.ca

sage: E.q_eigenform(10)

## Isogeny matrix

sage: E.isogeny_class().matrix()

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.